In the following discussion, we will treat finite-length sequences of length \(KL\)  as one period of a periodic sequence with period \(KL\) . Following the notation in [1]}, we use \(Z_x^{(L,K)} \in \mathbb {C^{Z\times Z}}\)  to denote the DZT of a sequence \(x\in \mathbb {C}^\mathbb {Z}\)  with period \(KL\) . The DZT of \(x\)  is then defined as [1]}
\(Z_x^{(L,K)}[n,k] \triangleq \frac{1}{\sqrt{K}}\sum _{l = 0}^{K-1}\underbrace{x[n+lL]}_{[l]}e^{-j2\pi \frac{k}{K}l},\quad n,k\in \mathbb {Z}.\) 
